Unit 3: Dynamics — Long Questions

Concept of force A force is commonly defined as a push or pull that starts, stops, or changes the magnitude and direction of an object's velocity.

Explanation We encounter many forces in our daily lives, some of which we apply to other objects and some that act on us.

i. For example, when we open a door, we apply force by pushing or pulling it.

ii. When sitting in a car that turns around a corner, we push against the seat due to the motion of the car.

iii. Force transfers energy to an object. For instance, when a man moves a wheelbarrow with a load, he first applies force to lift it and then applies force to push it. He adjusts the amount of force on each handle when turning to keep the wheelbarrow from tipping over.

iv. Other examples of forces acting on us include the downward force of gravity and the force of friction, which helps us walk on the ground.

i. Contact Forces

Contact forces are forces exerted by one object on another at the point of contact. Some examples of contact forces include:

(a) Friction:
Friction resists motion when the surface of one object comes into contact with another. For example, friction helps us walk by providing grip between our feet and the ground.

(b) Drag:
The drag is a resistant force caused by the motion of a body through a fluid. It acts opposite to the relative motion of the object through the surrounding fluid.

(c) Thrust:
It is an upward force exerted by a liquid on an object immersed in it. When we immerse an object in water, we feel an upward force. This force increases as we push the object deeper. For instance, a ship floats because this force balances its weight.

(d) Normal Force:
It is the force of reaction exerted by the surface on an object lying on it. This force acts outward and perpendicular to the surface. It is also called the support force of the object.

(e) Air Resistance:
It is the resistance (opposition) offered by air when an object falls through it.

(f) Tension Force:
It is the force experienced by a rope when pulled by a person or load pulls it.

(g) Elastic Force:
It is a force that brings certain materials back to their original shape after being deformed. Examples are rubber bands, springs, trampoline, etc.

ii. Non-Contact Forces

A non-contact force is defined as the force between objects that are not in physical contact. These forces work over a distance and are also known as field forces. Examples include:

(a) Gravitational Force:
This is an attractive force that exists among all bodies which have mass. For example, when an apple falls from a tree, it is pulled by the Earth's gravitational force. This force also keeps planets in their orbits around the Sun and holds the atmosphere and oceans on Earth. According to Newton's law of gravitation:

F = G (m1 m2) / r²

Where m1 and m2 are two masses, r is distance between masses and G is the constant of gravitational. Its value is 6.67 x 10^-11 Nm²kg^-2

(b) Electrostatic Force:
An electrostatic force acts between two charged objects. The Opposite charges attract each other and similar charges repel each other. Like gravitational force, electrostatic force is also a long-range force.

(c) Magnetic Force:
It is the force which a magnet exerts on other magnet and magnetic materials like iron, nickel, and cobalt. Magnetic forces can attract or repel, depending on the poles involved. For example, like poles repel, and unlike poles attract.

Strong and Weak Nuclear Forces:
These are also non-contact forces acting between the subatomic particles.

Fundamental Forces in Nature

There are four fundamental forces in nature. These are:
i. Gravitational Force ii. Electromagnetic Force iii. Strong Nuclear Force iv. Weak Nuclear Force

i. Gravitational Force

The gravitational force is the weakest among the four fundamental forces. It is a long-range force that extends to infinite distances, although it becomes weaker and weaker as the distance increases. This force is commonly observed in daily life, such as keeping planets in orbit and objects on Earth's surface.

ii. Electromagnetic Force

The electromagnetic force causes interactions between electrically charged particles. It includes both electrostatic and magnetic forces, which act through electromagnetic fields. These are long-range forces and are stronger than both gravitational and weak nuclear forces. This force is responsible for all chemical reactions, as it binds together atoms, molecules, and crystals. At the macroscopic level, electromagnetic force causes phenomena such as friction between surfaces in relative motion.

iii. Strong Nuclear Force

The strong nuclear force binds protons and neutrons together in the atomic nucleus, overcoming the repulsive electromagnetic force between positively charged protons. This is a short-range force, acting only within a distance of about 10^-14 meters. If the distance between nucleons increases beyond this range, the strong nuclear force ceases to act.

iv. Weak Nuclear Force

The weak nuclear force is responsible for the disintegration of atomic nuclei, such as during beta (β) decay. In this process, a neutron transforms into a proton, emitting a beta particle (electron) and an antineutrino. This force is stronger than the gravitational force but weaker than the electromagnetic force. It is a short-range force, acting over a distance of about 10^-17 meters.

Unification of Weak Nuclear and Electromagnetic Forces

A Pakistani scientist, Dr. Abdus Salam, along with Sheldon Glashow and Steven Weinberg, made significant contributions to the unification of the weak nuclear force and the electromagnetic force, forming what is known as the electroweak force. They were awarded the Nobel Prize in Physics in 1979 for their work.

Although these two forces appear different in everyday phenomena, the theory models them as two different aspects of the same fundamental force. This unification becomes apparent during interactions at very high energies. The effects of the electroweak force are observed in such high-energy interactions, demonstrating their common origin.

Free-Body Diagram

A free-body diagram is used to show the relative magnitudes and directions of all the forces acting on an object in a given situation. In other words, a free-body diagram is a special example of the vector diagrams.

Drawing a Free-Body Diagram

Suppose a book is pushed over the surface of a table. To represent the forces acting on the book using a free-body diagram:

• The object (the book) is represented by a box.

• The force arrows are drawn outward from the center of the box in the directions of the forces acting on the object.

• The length of the force arrows reflects the magnitude of the force, with longer arrows indicating greater force.

• The arrowhead shows the direction in which the force acts.

Each force is labeled to indicate the exact type of force, such as friction, gravity, normal force, drag, or tension in a string. These labels help identify the specific forces acting on the object.

Newton's First Law of Motion

Newton's First Law of Motion states that:
"A body continues its state of rest or of uniform motion in a straight line unless acted upon by some external force"
OR
A body continuous its state of rest or of uniform motion in a straight line provided no net force acts on it.

Example For example, a book placed on a table will remain stationary unless a force is applied to move it. Similarly, a ball rolling on the floor should continue to move with the same velocity in the absence of an applied force. However, in reality, the ball eventually stops due to friction, which acts as an opposing force.

Explanation When a fast-moving bus suddenly stops, the passengers tend to move forward because they want to continue their motion. On the other hand, when the bus starts moving quickly from rest, the passengers feel as if they are pushed back against the seat because their tendency is to retain their state of rest.

According to Newton's First Law, a bus moving on the road should continue its motion without any force exerted by the engine. However, in practice, if the engine stops, the bus comes to rest after covering some distance due to the friction between the tyres and the road. In outer space, if an object is thrown and no force acts upon it, it would continue to move indefinitely at a constant velocity.

Force according to Newton's First Law of motion

Newton's First Law also provides a definition of force:
"Force is an agency which changes or tends to change the state of rest or of uniform motion of a body." In simple words, force causes acceleration, meaning it can change the velocity of an object, either speeding it up, slowing it down, or changing its direction.

Inertia and First Law of Motion:
The property of a body to maintain its state of rest or uniform motion in a straight line is called inertia.

The resultant of all the forces acting on a body is called net force. A net force is required to change the velocity of an object.

Example When a net force is applied to a bicycle, it can pick up speed quickly. However, if the same force is applied to a truck, the change in motion may be less noticeable because the truck has more inertia.

Dependence Inertia is directly related to an object's mass. The greater the mass of an object, the greater its inertia. Therefore, the mass of an object is a measure of its inertia, that is why Newton's First Law is sometimes called the law of inertia.

Newton's second law deals with the acceleration produced in a body when a net force acts upon it. It states that:
"If a net external force acts upon a body, it accelerates the body in the direction of force. The magnitude of acceleration is directly proportional to the magnitude of force and is inversely proportional to the mass of the body"

Mathematical Expression

The second law can be expressed mathematically as:
a∝F
a∝1/m
a=constant×F/m

According to SI units, if m = 1 kg, a = 1 m/s², and F = 1 N, then the value of the constant is 1. Therefore, the equation simplifies to:
a = F/m
or F=ma

Newton OR Unit of force:
One newton is the force required to produce an acceleration of 1 m/s² in a body of mass 1 kg.
1N=1kg·m/s²

Effect of Force on Velocity:
Newton's second law also tells us that a force can change the velocity of a body by producing acceleration or deceleration in it. As velocity is a vector quantity, the change may occur in its magnitude, direction, or in both of them.

Newton's third law of motion states:
"For every action, there is always an equal but opposite reaction"

This means that whenever two bodies interact, if body A exerts a force on body B, body B exerts a force on body A that is equal in magnitude but opposite in direction. The forces are simultaneous and occur between two interacting bodies.

Action and Reaction Forces

For example, when we press a spring, the force exerted by our hand on the spring is called the action force. In response to this action, our hand experiences a force exerted by the spring, which is the reaction force (as shown in Fig. 3.10).

Action and Reaction Forces Never Balance Each Other

Action and reaction forces do not balance each other because they act on two different bodies. Although they are equal in magnitude and opposite in direction, they cannot cancel each other out because they are applied to different objects. Therefore, they never balance each other.

Forces Act in Pairs

Forces act in pairs when two objects interact, as stated in Newton's Third Law of Motion. According to this law, for every action, there is always an equal but opposite reaction. When two objects interact, one object exerts a force on the other, which is the action force, and the second object exerts a force back on the first object, which is the reaction force. While we often notice the action force that makes something happen, the reaction force involved is usually less apparent.

Examples

(i) Block on a Table:
Consider a block lying on a table as shown in Fig. 3.11. The force acting downward on the block is the weight. The block exerts a downward force on the table equal to its weight w. The table also exerts a reaction force F_N on the block. The two forces on the block balance each other and the block remain at rest.

(ii) Bullet Fired from a Gun:
When a bullet is fired from a gun, the bullet moves in the forward direction with a force F. This is the force of action. The gun recoils in the backward direction with a reaction force R (Fig.3.12).

Limitations of Newton's Laws of Motion

Newton's laws of motion can be applied with a very high degree of accuracy to the motion of objects and velocities which we come across in everyday life. However, problems arise when dealing with the motion of elementary particles having velocities close to that of light. For that purpose, relativistic mechanics, developed by Albert Einstein, is applicable. This theory helps in explaining and addressing the behavior of objects at speeds close to the speed of light.

Conclusion

After all this discussion, we can say that Newton's laws of motion are not exact for all types of motion, but they provide a good approximation unless an object is small enough or moving close to the speed of light.

Mass - Weight Comparison Table:

Mass | Weight

The characteristic of a body which determines the magnitude of acceleration produced when a certain force acts upon it is known as mass of the body. | The weight of an object is equal to the force with which the Earth attracts the body towards its center.

It is a scalar quantity. | It is a vector quantity, directed downward towards the center of the Earth.

It remains the same everywhere. | It varies from place to place depending upon the value of g.

The SI unit of mass is kilogram (kg). | The SI unit of weight is newton (N).

Practically, mass is measured by an ordinary balance. | Weight is measured using a spring balance.

Formula of mass is m = F/a | Formula of weight is w=mg

Balance Scales

Balance scales are commonly used to compare masses of objects or to weigh objects by balancing them with standard masses.

Mechanical Balances

A mechanical balance consists of a rigid horizontal beam that oscillates on a central knife-edge as a fulcrum. It has two end knife-edges equidistant from the center. Two pans are hung from bearings on the end knife-edges (Fig. 3.13). The material to be weighed is put in one pan, and standard weights are put on the other pan. The deflection of the balance may be indicated by a pointer attached to the beam. The weights on the pan are adjusted to bring the beam to equilibrium.

Mechanical Platform Balances

There is another type of mechanical balance used to weigh heavy items like flour bags, cement bags, and steel bars. These are called mechanical platform balances (Fig. 3.14). Standard weights are not required to use this balance. The reason is that the fulcrum of the beam of such a balance is kept very near to one end. Therefore, much smaller weights have to be put at the other end of the beam to bring it to equilibrium. These smaller weights have already been calibrated to the standard weights.

No standard weights are required to use in an electronic balance (Fig. 3.15). Only it has to be connected to a power supply. There are some models which can operate by using dry cell batteries. An electronic balance is more precise than a mechanical balance. When an object is placed on it, its mass is displayed on its screen. Nowadays, electronic balances also display the total price of the material if the rate per kg is fed to the balance.

Force Meter

A force meter is a scientific instrument that measures force. It is also called a newton meter or a spring balance (Fig. 3.16). Nowadays, digital force meters are also available. You have already learnt about mechanical and electronic balances. They measure the mass of objects in kilograms or its multiples. On the other hand, a force meter measures force directly in newtons (N).

Components and Working Principle

An ordinary force meter has a spring inside it. The upper end of the spring is attached to a handle. A hook is attached to the lower end of the spring that holds the object. A pointer is also attached to the spring at its upper end. A scale in newtons is provided along the spring such that the pointer coincides with zero on the scale when nothing is hung on the hook.

Measurement Process

The object to be weighed is hung on the hook. The mass of the object causes the spring to compress. The pointer indicates the weight of the object. However, some force meters are also based on the stretching of the spring when a load is hung. In this case, the pointer is attached at the lower end of the spring. In some spring balances, the scale measures the mass, which can be readily converted into newtons by multiplying the mass in kg with the value of g=10ms^-2.

Digital Force Meters

A digital force meter measures directly the weight of the object in Newtons (Fig. 3.17).

Friction

"The force that opposes the motion of moving objects is called friction."

Explanation

When a cricket ball is hit by the bat, it moves on the ground with a reasonably large velocity. According to Newton's first law of motion, it should continue to move with constant velocity. But practically, we observe that it eventually stops after covering some distance. It is the force of friction between the ball and the ground that opposes the motion of the ball.

Dissipative Effect of Friction

Friction is a dissipative force due to which the energy is wasted in doing work to overcome friction. The lost energy appears in the form of heat.

Example

A very common example of energy dissipation is the rubbing of hands (Fig. 3.18). When we rub our hands, heat is produced due to friction, and our hands become warm. Similarly, the temperature of machines rises due to friction between its moving parts, which can cause many problems. The tyres of vehicles also wear out after becoming too hot due to friction between tyres and the road. Shooting stars seen in the sky at night also happen due to friction of air. These are actually asteroids that enter the Earth's atmosphere. As they are moving, air resistance causes the generation of heat. Their temperature becomes so high that they start burning and ultimately disintegrate.

Sliding Friction

The friction between two solid surfaces is called sliding friction, which can be divided into two categories:
i. Static Friction
ii. Kinetic Friction

i. Static Friction

Static friction is the opposing force that prevents an object from starting to move when it is at rest on a surface.

Example

Let us consider the motion of a block on a horizontal surface. The arrangement is shown in Fig. 3.19. When a weight is put in the pan, a force F=T equal to the sum of this weight and the weight of the pan acts on the block. This force tends to pull the block. At the same time, an opposing force appears that does not let the block move. This opposing force is the static friction Fs.

ii. Kinetic Friction

If we go on adding more weights in the pan one by one in small steps, a stage will come when the block starts sliding on the horizontal surface. This is the limit of static friction still exists. It is known as kinetic friction

Rolling Friction

"The force of friction between a rolling body and the surface over which it rolls is called rolling friction."

The idea of rolling friction is associated with the concept of a wheel. In our everyday life, we observe that a body with wheels faces less friction as compared to a body of the same size without wheels. Ball bearings also play the same role as is played by the wheels. Many machines use ball bearings to greatly reduce friction. The rolling friction is about one hundred times smaller than the sliding friction.

The reason for rolling friction to be less than sliding friction is that there is no relative motion between the wheel and the surface over which it rolls. The wheel touches the surface only at a point; it does not slide.

Terminal Velocity

Terminal velocity is the constant velocity achieved by an object when the upward force of air resistance balances the downward force of gravity, causing the object to stop accelerating and continue falling at a constant speed.

Explanation

When an object falls freely under the influence of gravity, it is initially accelerated by g=10 ms^-2. However, in reality, air resistance affects the motion of falling objects. If we drop two objects of the same weight, such as a cricket ball and a piece of Styrofoam, from a certain height, they would hit the ground at the same time only in the absence of air resistance. In practice, the cricket ball, with a smaller surface area, would fall faster than the Styrofoam, which faces greater air resistance due to its larger surface area. As the object falls faster, the air resistance increases. Eventually, a point is reached where the upward air resistance force equals the downward force of gravity. At this stage, the object stops accelerating and continues to fall at a constant speed—this speed is known as the terminal velocity.

Applications

Metetorites Even heavy objects like meteorites do not gain infinite velocity as they fall towards Earth. They reach a terminal velocity before hitting the ground due to air resistance.

Paratroopers The principle of terminal velocity is important for paratroopers. The large surface area of a parachute increases air resistance, allowing for a controlled and safer downward movement by reducing the falling speed to a safe terminal velocity.

The following methods are used to reduce friction:

i. Polishing Sliding Parts

The parts which slide against each other are highly polished to reduce surface roughness, which in turn reduces friction.

ii. Using Oil or Grease

Since the friction of liquids is less than that of solid surfaces, oil or grease is applied between the moving parts of machinery to reduce friction.

iii. Using Ball Bearings

As rolling friction is much less than sliding friction, sliding friction is converted into rolling friction by the use of ball bearings in machines and wheels under heavy objects.

iv. Streamlining Vehicles

High-speed vehicles, airplanes, and ships also face friction while moving through air or water. If the front of a vehicle is flat, it faces more resistance from air or water. Therefore, bodies moving through air or water are streamlined to minimize air or water friction. In this case, the air passes smoothly over the slanting surface of the vehicle. This type of flow of air is known as streamline flow. Vehicles designed to be pointed from the front are said to be streamlined.

Momentum

Momentum is defined as the quantity of motion of a moving body. It depends on the mass and velocity of the object. The greater the mass, the greater the momentum; similarly, the greater the velocity, the greater the momentum. The momentum of a moving body is the product of its mass and velocity. Mathematically, this is represented as:
P=m×v...................... (1)

Momentum is a vector quantity, meaning it has both magnitude and direction. The SI unit of momentum is kg·m/s (kilogram meter per second), which can also be written as N·s (Newton second).

Impulse

Impulse refers to the total change in momentum of an object when a large force acts on it over a short interval of time. In practical situations, such as when a ball is hit by a bat, the exact value of the force and the time duration are difficult to measure. However, the change in momentum (initial and final velocities) can be easily determined. The average acceleration during the time interval Δt is given by:
a= Δv/Δt = (vf-vi)/Δt ................................ (2)

According to Newton's second law of motion, the average force (F) during this time interval is:
F= ma = m(Δv/Δt)

Or F×Δt=m(Δv)=m(vf-vi)................................ (3)

Equation (3) shows that F and Δ t cannot be exactly known but their product which is equal to the change of momentum (mvf - mvi) can be calculated. For such cases, the product F × Δt is called as Impulse of the force.

When a large force F acts on an object for a short interval of time, the impulse of the force is defined as the total change in momentum of the object.

By dividing both sides of equation (3) by Δt, we can express force in terms of momentum:
F = m(Δv)/Δt ...................................... (4)

where m(Δv) is the change in momentum Δp. Equation (4) gives the value of force in terms of momentum i.e., force acting on an object is equal to the change in momentum of the object per unit time.
F = Δp/Δt

This equation shows that force is equal to the rate of change of momentum. Thus, Newton's second law of motion can be rewritten as:
"The rate of change of momentum of a body is equal to the force acting on it."

The direction of the change in momentum is the same as the direction of the force.

A system is a collection of objects. If no external force acts on any object within the system, it is called an isolated system.

Explanation

Consider a system of two balls with masses m₁ and m₂, moving along a straight line in the same direction with velocities v₁ and v₂ respectively. If v₁>v₂, the balls will eventually collide. Let their velocities after the collision be v₁' and v₂'.

Total Momentum Before Collision=m₁v₁+m₂v₂
Total Momentum After Collision = m₁v₁'+m₂v₂'

The principle of conservation of momentum states:
"If no external force acts on an isolated system, the total momentum of the system remains constant before and after the collision."

Mathematically,
m₁v₁+m₂v₂= m₁v₁'+ m₂v₂'

Example Collision of Two Identical Balls

Consider two identical balls as shown in diagram, where the first ball is moving with velocity v₁ and the second ball is initially at rest (v₂=0). When the moving ball collides with the stationary ball, the momentum is transferred from the first ball to the second. The first ball slows down and comes to rest. The second ball gains momentum and starts moving with the velocity that the first ball had before the collision.

The total momentum of the system before collision is the momentum of the first ball (m₁v₁). After the collision, the momentum of the second ball (m₂v₂') equals the momentum lost by the first ball. Therefore, the total momentum remains the same.

The principle of conservation of momentum is not limited to large objects but also applies to micro-objects such as atoms and molecules. This universal principle helps explain various physical phenomena, from everyday collisions to complex atomic interactions.